I like
@Oni_Rinku's angle of making it a pre-Hogwarts thing, but I would add the following caveat:
This girl, on account of being about 50% Magical Creature, is plugged into the Magical World and knows that Harry is the Boy-Who-Lived; but
doesn't know that he's basically a muggle-born who doesn't know about Magic. She
wrongly assumes he is in on The Masquerade.
Probably I would have it that her parents went into hiding by living as Muggles during the Voldemort Times, and they were like "this is actually kinda nice", so they continue to live mostly as Muggles, just going to visit relatives in the Magical World or whatever. Mostly that would be to explain why this girl would also be living on Privet Drive, or in Muggle Surrey more generally.
So I'm seeing a scenario where she has her first real experience with Fantastic Racism, and she's feeling bad about it, and is walking around outside or something, and runs into the Boy-Who-Lived, and they start talking, and he comes off as this completely enlightened post-racist kid... because he literally doesn't even know about the differences he's "supposed" to be racist about. (The fact her parents are living like Muggles also means that she would still have Muggle Common Sense, so she wouldn't come off as weirdly helpless around cars or whatever.)
So they start hanging out.
I would preserve the misunderstanding where she thinks he knows about Magic, and meanwhile he would think she was a Chuunibyou if he knew the word for it, but Harry's home life makes him really, really empathetic about wanting to pretend about that. And anyway he's so starved for friendship that he is willing to uncritically accept a
lot of strangeness. I would mine that for dramatic tension and comedy, where they are having One Dialogue Two Conversations a lot.
There are a couple of different ways that could go, just like, story events that could happen, with some overlap:
1) Harry learns that Magic is Actually A Real Thing from her, and she becomes his guide to the Magical World. So he would learn about his parents and all that, but the pure wonder of the magical world would be tainted because everything he gets is from somebody from the Underclass.
2) He doesn't learn about the Magical World from her, but instead when he gets his Hogwarts letter, and he quietly decides to pretend like he actually knew all along. Maybe a few years later he admits he originally just thought she was a Chuuni, and they have a laugh.
3) She admits that she's a Succubus or whatever, and he admits that he can talk to snakes. She's really flattered that he would respond to her confession of being a "suspicious" Magical Creature by sharing the secret of his Dark Talent and trusting her; meanwhile, he's still ignorant of the Magical World generally and is just reciprocating the secret of his "freakishness" when she tells him hers.
4) They attend Hogwarts together and she's the Designated Heroine.
5) Her parents have her attend Muggle School so she can live in the Muggle World and have a good job and not have to deal with the Fantastic Racism; so he continues to only see her during the summer. I would take this story as being almost a series of "interludes" where instead Harry only refers to the events of the School Year. I would probably have his friendship with this girl cause increasingly large butterflies to the plot which grow over time.
6) Her dad works at Grunnings and is really leery about her hanging out with that asshole Vernon's kid, and then does a double-take when he finds out that actually that boy is The Boy Who Lived.
7) Her dad works at Grunnings and hears a lot of stories from Vernon about "his ungrateful nephew", and although he took it with a grain of salt, he's still leery about it and thinks it's weird how opposite-world things look from the outside. Also yeah, The Boy Who Lived thing happens.
8) Her parents unsubtly push them together; they are cynically trying to make The Boy Who Lived into a Creature Rights Activist, and she is mad they're trying to exploit her friendship, but also grateful for the chance.
9) This girl's presence acts as a catalyst for Dudley's emotional growth which mostly happened off-screen in the books and culminated in that discussion in book seven; so Dudley grows up faster in this story, and reconciles with Harry earlier and more fully. Heck if Smeltings was co-ed (I dunno if it was in canon but Make It So for this story), you could even go full Romcom and have a bit of a love triangle between her, Harry, and Dudley, where she goes to school with Dudders and sees him grow up? Or at least use it that Harry can be jealous about Dudley getting to see her during the school year and not him.
Anyway,
I would probably write this from
her perspective.
Like, I would mine the dramatic irony that she's thinking that the Boy-Who-Lived is so much more humble and kind than she expected, while making it clear to the Reader that she's missing that he doesn't know anything about the Magical World. Possibly by only having short interludes from Harry's perspective, like Point Five above.
And it allows you to completely avoid having Hogwarts as a setting, which I think would be interesting in an HP fic because it's different. So going from Point Four above, she meets up with him every summer, and she has a normal, adventure-free childhood.
Another option, maybe running with Point Nine, would be that they write each other a bunch of letters? A story that's just a series of letters between two people going back and forth, that's a solid, well-established premise, but I dunno if it's something I could write. Heck you could mix-and-match, have chapters that are just letters during the school year, interwoven with narrative chapters during the summer.
That's about what comes to mind from this premise.
